Solve the following equations for x where 0 ≤ x < 2π.
tan(x)sin(2x) = √3 sin (x)
and
cos(2x)−cos(x) = −sin2(x)+1/4
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Stuck on these particular types. Have no idea how to start.
tan(x)sin(2x) = √3 sin (x)
tanx(2sinxcosx) = √3sinx
divide by sinx
2tanxcosx = √3
2(sinx/cosx)(cosx) = √3
sinx = √3/2
I recognize x as one of the special angles, x = 60°
but sine is also positive in quadrant II
so x = 180-60 = 120°
x = 60° or x = 120°
for the second, I am confused by your notation.
at the front you have cos(2x) , no problem there, but in
sin2(x) , did you mean sin(2x) or sin^2 x ?
Apologies, I meant" -((sin^2)x + 1/4) "
To solve the equation tan(x)sin(2x) = √3 sin(x), we can begin by simplifying the equation using trigonometric identities.
Step 1: Simplify the equation tan(x)sin(2x) = √3 sin(x).
First, we rewrite tan(x) as sin(x)/cos(x) and sin(2x) as 2sin(x)cos(x).
The equation becomes: (sin(x)/cos(x))(2sin(x)cos(x)) = √3sin(x).
Step 2: Simplify further.
Now, we can cancel out sin(x) from both sides of the equation. If sin(x) ≠ 0, we can divide both sides by sin(x). To ensure we don't divide by zero, we need to consider sin(x) = 0 as a possible solution later.
Dividing both sides by sin(x), the equation becomes: 2cos(x) = √3.
Step 3: Solve for cos(x).
To solve for cos(x), divide both sides of the equation by 2: cos(x) = √3/2.
Step 4: Find the angle x that satisfies cos(x) = √3/2.
Using the unit circle or trigonometric values, we can find the angles where cos(x) = √3/2. In the specified range 0 ≤ x < 2π, the solutions for x are π/6 and 11π/6.
Therefore, the solutions to the equation tan(x)sin(2x) = √3 sin(x) in the specified range are x = π/6 and x = 11π/6.
Moving on to the second equation:
Step 1: Simplify the equation cos(2x) − cos(x) = −sin^2(x) + 1/4.
Using the double angle formula, we can rewrite cos(2x) as 2cos^2(x) - 1.
The equation becomes: 2cos^2(x) - 1 − cos(x) = −sin^2(x) + 1/4.
Step 2: Rearrange the equation.
Rearranging the equation, we have: 2cos^2(x) − cos(x) + sin^2(x) = 5/4.
Step 3: Apply the Pythagorean identity.
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute 1 − cos^2(x) for sin^2(x) in the equation:
2cos^2(x) − cos(x) + 1 − cos^2(x) = 5/4.
Step 4: Simplify the equation.
Combining like terms, we get: cos^2(x) − cos(x) + 1/4 = 0.
Step 5: Solve the quadratic equation.
We can factor the quadratic equation as follows: (cos(x) − 1/2)(cos(x) − 1/2) = 0.
This gives us two possible solutions: cos(x) = 1/2 and cos(x) = 1/2.
Step 6: Find the angles that satisfy cos(x) = 1/2.
Using the unit circle or trigonometric values, we can find the angles where cos(x) = 1/2. In the specified range 0 ≤ x < 2π, the solutions for x are π/3 and 5π/3.
Therefore, the solutions to the equation cos(2x) − cos(x) = −sin^2(x) + 1/4 in the specified range are x = π/3 and x = 5π/3.
To solve these equations, we'll go step by step for each equation. Let's start with the first one:
Equation 1: tan(x)sin(2x) = √3 sin(x)
Step 1: Simplify the equation if possible:
Since tan(x) = sin(x)/cos(x), we can rewrite the equation as:
(sin(x)/cos(x)) * sin(2x) = √3 sin(x)
Step 2: Multiply both sides by cos(x) to eliminate the denominator:
sin(x) * sin(2x) = √3 sin(x) * cos(x)
Step 3: Expand the product on both sides:
sin(x) * (2sin(x)cos(x)) = √3 sin(x) * cos(x)
Step 4: Cancel out sin(x) on both sides (Note: be careful with dividing by sin(x) since it might be zero at certain values of x):
2sin(x)cos(x) = √3 cos(x)
Step 5: Divide both sides by cos(x) (again, be mindful of dividing by cos(x) since it might be zero at certain values of x):
2sin(x) = √3
Step 6: Divide both sides by 2 to isolate sin(x):
sin(x) = √3/2
Step 7: Solve for x using the inverse of the sine function (sin^(-1)):
x = sin^(-1)(√3/2)
Since we are given the constraints 0 ≤ x < 2π, the solution for x is:
x = π/3 (60 degrees)
Now, let's move on to the second equation:
Equation 2: cos(2x)−cos(x) = −sin^2(x) + 1/4
Step 1: Simplify the equation by expanding the double angle identity:
2cos^2(x) - 1 - cos(x) = -sin^2(x) + 1/4
Step 2: Rearrange the terms to put the equation in a quadratic form:
2cos^2(x) - cos(x) + (sin^2(x) - 5/4) = 0
Step 3: Combine like terms on the left side:
2cos^2(x) - cos(x) + (1 - cos^2(x) - 5/4) = 0
Step 4: Simplify further:
2cos^2(x) - cos(x) + (1 - cos^2(x) - 5/4) = 0
Step 5: Combine like terms:
cos^2(x) - cos(x) - 1/4 = 0
Step 6: Factor the quadratic equation or use the quadratic formula to solve for cos(x). Let's use factoring:
(cos(x) - 1/2)(cos(x) + 1/4) = 0
Step 7: Set each factor equal to zero and solve for cos(x):
cos(x) - 1/2 = 0 or cos(x) + 1/4 = 0
Solving each equation separately:
For cos(x) - 1/2 = 0, we have cos(x) = 1/2. This occurs when x = π/3 (60 degrees) or x = 5π/3 (300 degrees).
For cos(x) + 1/4 = 0, we have cos(x) = -1/4. This doesn't have a solution within the given constraints 0 ≤ x < 2π.
Therefore, the solutions for the second equation within the given constraints are:
x = π/3 (60 degrees) and x = 5π/3 (300 degrees).